In the field of condensed matter physics, extensive efforts have been put into the exploration of the anomalous Hall effect (AHE) since the dominance of Berry phase curvature and spin-orbit-coupling-mediated scattering provides a powerful probe for topological order in quantum materials, a key pursuit for developing next-generation dissipationless spintronic devices. Nonetheless, the underlying mechanisms of the AHE in the low conductive dirty regime (longitudinal conductivity ${\sigma }_{xx}<{10}^4{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$) remain insufficiently understood because of the difficulty in preparing pure ferromagnetic metals with low conductivity. To address this, we utilized the technology of low-energy cluster beam deposition to control the structural disorder and successfully prepared ferromagnetic nanogranular CoFe films with a low conductive metallic state, which allow us to systematically investigate the AHE of pure ferromagnetic metals in the dirty regime. Our results reveal a nonmonotonic evolution of anomalous Hall conductivity ${\sigma }_{\mathrm{AHE}}$ with increasing cluster size. The possible dominant mechanism is the intrinsic contribution derived from the scaling relationship between ${\sigma }_{xx}$ and ${\sigma }_{\mathrm{AHE}}$ notwithstanding the presence of the extrinsic skew scattering. More importantly, ${\sigma }_{\mathrm{AHE}}$ not only falls within the dirty regime but also is consistent with the widely established scaling exponent of ${\sigma }_{\mathrm{AHE}}\propto {\sigma }_{xx}^{1.6}$, which indicates that the degenerative AHE arises from the damping of the intrinsic contribution. In this work, we show the potential of cluster-assembled engineering in the exploration of the quantum transport properties for future electronics devices.

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• Anomalous Hall effect
• Magnetism
• Magnetotransport
• Spintronics
• Granular solids
• Thin films
• Transition metals
• Resistivity measurements